Makalah Osilator Harmonik

 

 Dntn Iueona Dhincoin F]OEN[FW ANWDFCOI

Kosusuc Feha 2 Kosusuc Feha 2 @usronctn

1>?:>080808. Entnr Gheninc` Gheninc` .......... ............... ......... ......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ......... .........> .....> >.7. Wudusnc Wudusnc Dnsnena .... ......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ......... ......... .......... ..........> .....> >.=.

[ubunc [ubunc .......... ............... .......... .......... ......... ......... .......... .......... .......... .......... .......... .......... .......... .......... .......... .......... ............. ...........> ...>

GNG OO \HDGNAN]NC 7.> Fsoentfr Anrdfcoi .............................................................................. ..............................................................................7 7 7.7 Fsoentfr Eochnr knc Cfc Eochnr ..........................................................8 ..........................................................8  

7.7.>. Fsoentfr Eochnr......................................... Eochnr..........................................................................8 .................................8

 

7.7.7. Fsoentfr Cfc Eochnr........................................ Eochnr................................................................... ...........................8 8

7.= Fsoentfr Anrdfcoi ]hkhrancn.............................................................: 7.; Hchr`o Fsoentfr Anrdfcoi ]hkhrancn..................................................1 ]hkhrancn..................................................1 7.8 \hrsndnnc @hrni Fsoenso Whkndnc.....................................................>< Whkndnc.....................................................>< GNG OOO \HCX[X\ =.>. Ihsodpuenc .............................. .................................................... ............................................ .................................... .............. 7< KNJ[NW \X][NIN ............................................ .................................................................. .......................................... .................... 7>

ooo

 

GNG O \HCKNAXEXNC >.>.

Entnr Gheninc`

@hrni anrdfcoi shkhrancn nknena `hrni gfeni - gneoi ghckn dheneuo suntu totoi ihshodgnc`nc thrthctu khc`nc gncynicyn `htnrnc ghckn knend shtonp shifc sheneu ifcstnc. @hrni Anrdfcoi ]hkhrancn nknena `hrni gfeni gneoi shmnrn thrntur  dheneuo totoi ihshodgnc`nccyn khc`nc gncynicyn `htnrnc ghckn knend shtonp shifc sheneu sndn ntnu ifcstnc. ]htonp `hrni ync` thrbnko shmnrn ghruenc` knend shenc` wnitu ync` sndn koshgut `hrni phrofkoi. Inrhcn `hrni oco thrbnko shmnrn thrnt thr ntur ur dnin dnin kosh koshgu gutt bu`n bu`n sh shgn gn`n `noo `h `hrni rni an anrd rdfc fcoi oi.. Npng Npngoe oenn sunt suntu u pn pnrt rtoi oihe he dheniuinc `hrni phrofkoi pnkn eoctnsnc ync` sndn dnin `hrnicyn koshgut `hrni  fsoenso/`htnrnc. Ghctui ync` shkhrancn knro `hrni phrofkoi nknena ghckn ync`  ghrfsoenso pnkn ubuc` ph`ns inrhcncyn iotn dhcyhgutcyn `hrni anrdfcos shkhrancn, feha inrhcn otu ihefdpfi indo ninc dhcyusuc dninena ync` ghrbukue ―Fsoentfr Anrdfcoi—.

>.7.

Wudusnc Dn Dnsnena

>. Gn`nod Gn`nodncn ncn ync` ync` kodnisu kodnisuk k fsoentf fsoentfrr anrdfcoi anrdfcoi66 7. Npn snbn snbn npeoinso npeoinso knro fsoentfr fsoentfr anrdfcoi anrdfcoi knend knend ihaokup ihaokupnc nc shanro-anro6 shanro-anro6

>.=.

[ubunc \hcueosnc

>. Xctui dhc`htnauo dhc`htnauo ync` kodnisuk kodnisuk fsoentf fsoentfrr anrdfcoi. anrdfcoi. 7. Xctu Xctui i dhc` dhc`ht htna nauo uo np npeo eoin inso so kn knro ro fs fsoe oent ntfr fr an anrd rdfc fcoi oi kn knen end d ih ihao aoku kupn pnc c shanroanro.

>

 

GNG OO \HDGNAN]NC

Boin shguna sosthd knend ihshtodgnc`nc stngoe stntos ntnu kocndos, ihtoin sosthd thrshgut kopocknainc shkoiot knro pfsoso ihshtodgnc`nc, `hrni fsoenso ync` koansoeinc koshgut `hrni anrdfcoi. @hrninc shphrto oco shroc` thrbnko ko nend knc kosheokoio, kosheok oio, gnoi knro prnitos shrtn sukut pncknc` pncknc` thfrotos, knend josoin knc thicoi. Ghghrnpn mfctfa knro `hrninc thrshgut nknena ph`ns henstos, gnefi ync` ehctur,  phckueud ync` ghr`htnr, rhsfcncso knro rfc``n uknrn, knc `hrninc duntnc knend rnc`inonc eostroi. Xctui Xc tui dhduen dhdueno, o, iotn iotn ninc ninc dhdphe dhdphenbnr nbnroo `hrni `hrni fsoent fsoentfr fr anrdfc anrdfcoi oi eocohr  eocohr  (`hrninc ync` koansoeinc knro phrpocknanc ihmoe knro sosthd ihshtodgnc`nccyn) kn knen end d sntu sntu kodh kodhcs cso. o. [hra [hraoc ockn knri rinc nc oc ocie ieus usoo `h `hshi shinc nc kn knen end d `h `hrn rnin inc c th thrsh rshgu gutt dhc`nrna pnkn phcyheokoinc fsoentfr anrdfcoi thrhknd. Xctui dhcbn`n `hrni  fsoenso knend `hshinc, ghghrnpn `nyn histhrcne anrus kothrnpinc. ]osthd ghrfsoenso shphrto oco koshgut fsoentfr pnisn ntnu kokfrfc`. Ihtoin phrpocknanc sosthd knro ihshtod ihsh todgnc gnc`nc `nc ghsnr, ghsnr, sosthd sosthd oco tokni tokni en`o en`o eocohr. eocohr. ]osthd ]osthd ghrfso ghrfsoenso enso shphrto shphrto oco koshgut cfc eocohr. Gng oco thrutndn kotubuinc uctui dhdphenbnro sosthd eocohr  thrdnsui fsoentfr thrhknd knc fsoentfr anrdfcoi kopnisn. ]tuko thctnc` fsoenso cfc eocohr ninc kosheokoio knend gng oco. ]hmnrn udud, fsoenso sosthd ync` thrbnko ko nend nknena fsoenso cfc eocohr 

7. 7.>. >.

Fsoe Fsoent ntfr fr eoco eocohr hr knc knc Cfc Cfc Eoco Eocohr hr

\hrtodgnc`inc shguna pnrtoihe ghrdnssn d ghr`hrni knend dhknc `nyn if ifcs cshrv hrvnt ntoj oj kh khc` c`nc nc hc hchr hr`o `o pf pfth thcs cson onee Z(x) Z(x) kn knro ro pn pnrt rtoi oihe he shgn` shgn`no no ju juc` c`so so kn knro ro  phrpocknanc ync` ko`ndgnrinc feha iurvn, shphrto ync` kotucbuiinc pnkn `ndgnr  kognwna oco.

@ndgnr 7.>.

7

 

@ndg @n dgnr nr kont kontns ns dhcu dhcucb cbui uiin inc c shgun shguna a pn pnrto rtoih ihee d kn knc c hc hchr` hr`oo H sh shkn knc` c`  ghr`hrni knend juc`so hchr`o pfthcsone Z(x) kotucbuiinc khc`nc `nros thgne. Iurvn ync` putus-putus nknena juc`so pfthcsone pnrngfen. Xctui dhknc `nyn ifcshrvntoj, hchr`o tftne H knro pnrtoihe nknena

 H4 I + Z 4ifcs tnc

(7.>)

Boin x Boin  x nknena  nknena ihmhpntnc knro pnrtoihe, >

7

 H4 dx +Z ( x ) 7

(7.7)

Kodncn boin iotn shehsnoinc knend x knend x dhc`ansoeinc  dhc`ansoeinc

 x 4

−

 7 kx  4¾  Y  HH ∗Z ( x ) T kt  d

(7.=)

Boin H 4 H. [otoi x> knc x7  koshgut koshgut totoi gneoi, knc ihmhpntnc  pnrtoihe pnkn totoi-totoi thrshgut nknena cfe. [otoi-totoi oco kophrfeha khc`nc dhcyhehsnoinc Ht ” Z(x) 4  knc x7 ihtoin hchr`o ehgoa  ghsnr knro H 4

−7∩ d

 x 7

 x >

  kx

−  H ∗Z ( x )

(7.;)

]hknc`inc wnitu phrofkh [ knro sntu fsoenso nknena

[ 47 ( t 7 ∗t > ) 4 − 7 d

 x 7

∩

 x>

  kx −   HH ∗Z ( x )

=

(7.8)

 

\hrsndnnc (7.;) knc (7.8) tokni knpnt koshehsnoinc ihmuneo iotn dhc`htnauo  ghctui juc`so Z pfthcsone (x). @hrninc pnrtoihe pnkn knhrna ko eoc`iuc`nc x pfthcsone knpnt kotueos shgn`no

()

4 >/7 ( ∗

>)

Fsoe Fs oenso nso eoco eocohr hr kn knc c cf cfc c eoco eocohr hr Io Iotn tn dosn dosnei einc nc toto totoi i nsne nsne th threh rehtni tni ko toto totoi  i  ihshodgnc`nc shaoc``n x>), iotn dhcknpntinc

>

7

Z ( x )4 ix + 7

> ;

∁ x

;

+. . . . . (7.>7)

;

 

]henoc otu, inrhcn `hrninc pnrtoihe knend dhknc `nyn ifcshrvntoj, khc`nc dhc``ucnin dhc`` ucninc c khjocoso khjocoso

()

4 ∗

/ knc

dhc``nctoinc Z(x) knro phrsndnnc

(7.>7), iotn knpnt dhcueos () 4 ∗∗∁ = (7.>=)

7.>. 7.>.>. >. Fsoe Fsoent ntfr fr Eo Eoco cohr hr

Knend phckhintnc phrtndn, iotn knpnt dhc`ngnoinc shdun ane ihmuneo ync`  phrtndn ko phrsndnnc (7.>7) knc (7.>=) shaoc``n >

Z  (  ( x )4 ix 7 7

(7.>;)

 J ( x )4∗ix

(7.>8)

Kosoco

(  ) (  )

i 4

k 7 Z  kx 7

4

 x  x∗ x<

kJ  kx   xx∗ x <

(7.>:)

]hbni (k7Z/kx7  )< ghrcoeno pfsotoj, i bu`n ninc ghrcoeno pfsotoj. Feha inrhcn otu `nyn J(x) 4 - ix sheneu konrnainc dhcubu pusnt knc shgnckoc` khc`nc x. @nyn shphrto oco koshgut koshgut `nyn phdueoa eocohr. \fthcso ync` shsuno khc`nc khc`nc `nyn thrshgut nknena pnrngfen shphrto ync` koghroinc feha phrsndnnc (7.>;) knc kotucbuiinc feha iurvn putus-putus pnkn @ndgnr 7.> knc 7.7 uctui coeno ync` ghrghkn knro i.

7.>. 7.>.7. 7. Fsoe Fsoent ntfr fr Cfc Cfc Eoco Eocohr hr

\hrsndnnc (7.>7) knc (7.>=), `nyn tokni en`o eocohr inrhcn nkncyn ostoena x = , shdhctnrn pfthcso tokni en`o pnrngfen inrhcn nkncyn ostoena x ; . Ghrgn`no ghctui  `nyn knc pfthcsone ync` kooeustrnsoinc pnkn @ndgnr 7.7 uctui sosthd khc`nc  phrpocknanc ync` ghsnr (shaoc``n tokni nkn en`o eocohr). Knro phrsndnnc (7.>=) uctui sosthd cfc eocohr, ynotu

( )4∗ ix ∗ ∁ x

 J   x

7

(7.>0)

ync` dhrupninc budena ync` snc`nt ihmoe kognckoc`inc khc`nc Iotn anrus oc`nt i, cnduc ghsnrcyn ghsnrcyn knc tnckn dhdphc`nr dhdphc`nruao uao auguc`nc auguc`nc eocohr ” ix, `nyn ync` koansoeinc J(x). Boin . \h`ns ghrnkn pnkn totoi ihshtodgnc`nc x 4 < J(x)4-ix

(7.>)

J(x)4 `nyn phdueoa (I) i 4 ifctnctn ph`ns (C/d) x 4 sodpnc`nc ph`ns \hrsnd \hr sndnnc nnc kontns kontns koshgu koshgutt shgn`no shgn`no Auiud Auiud Affih. Affih. @nyn @nyn phdueo phdueoa a ync` ync`  ghihrbn pnkn ghckn shgnckoc` khc`nc sodpnc`nc x knro ph`ns ync` korhctnc`inc ntnu kothinc knro pfsoso shtorcgnc`cyn. \fsoso ph`ns ync` kþhctnc nc knc kothinc knro pfsthr ihshtçrgnc`nc knpnt keeoant pnkn `ndgnr 7.> knc 7.7.

:

 

@ndgnr 7.7. \h`ns kotnroi ih incnc knro totoi shtodgnc` \hrofkhh fsoentfr \hrofk fsoentfr anrdfcom anrdfcom shkhran shkhrancn cn thrcyntn thrcyntn ghr`nctuc` ghr`nctuc` pnkn thincnc ph`ns knc dnssn ync` ghrfsoenso khc`nc dhchrnpinc afiud F Chwtfc, ynotu 2

 έ J 4 d⃐ x

(7.7)

\hrsndn \hr sndnnc nc fsoens fsoensoo anrdfc anrdfcom om shkhran shkhrancn cn kophrf kophrfeha eha khc`nc khc`nc dhcsug dhcsugsto stotus tusoin oinc c  phrsndnnc 7.> knc 7.7. dhcbnko 2 J(x) 4 -ix 7

k  x d . 7   + ix 4< kt 

(7.=)

\hrsnd \hr sndnnc nnc (7.=) (7.=) dhrupn dhrupninc inc phrsnd phrsndnnc nnc kojjhr kojjhrhcso hcsone ne fsoent fsoentfr fr anrdfc anrdfcom om shkhran shkh rancn cn knc `hrnic `hrnicyn yn koshgu koshgutt `hrnin `hrninc c anrdfc anrdfcoi oi shkhra shkhrancn ncn.. \hcyhe \hcyhehsno hsnonc nc  phrsndnnc (7.=) 7

k  x ∗i   4   x 7 d kt   x

 4 ∩ kx  x  x <

−

∗i  d

 ∩ kt  ¶

Khc`nc dhdosneinc

−

 x ∗i    t   4  x < d

(7.;)

i  4 Ϗ7,phrsndnnc (7.;) dhcbnko d

  x 4 ∗Ϗ 7 t 7 ¶  x −  <

¶

 x 4 o 7 Ϗ 7 t 7  x <

¶

 x 4¾oϏt   x <

− 

 x 4 h¾oϏt   x<  x 4 x < h ¾oϏ

(7.8)

Ghctui enoc phrsndnnc (7.8) knpnt kotueosinc shgn`no ghroiut 2

 x 4 Nh + Gh∗oϏt  

(7.:)

0

 

\hrsndnnc (7.:) knpnt kotueosinc ih knend ghctui enoc ghroiut 2

 x 4 Nmfs ( Ϗt + ∈ )

(7.0)

\hrsn \hrsncu cunc nc (7.0) (7.0) rrhru rrhrupn pnin inc c sf sfeu euso so ph phrsn rsndn dnnc nc

an anrd rdfc fcoi oi

shkhr shkhran ancn cn..

Khc`nc N, h, &, knc rchrupninc ifcstnctn. N nknena ndpeotukf fsoenso knc x nknena nkn ena sodpnc sodpnc`nc `nc.. ]hknc` ]hknc`inc inc juc`so juc`so mfsocu mfsocuss (nt + h) koshgu koshgutt jnsh `hrni knc ifcstn ifc stnctn ctn nt koshgu koshgutt £fctnc £fctnctn tn jnsh ntnu sukut jnsh. jnsh. ]hendn ]hendn sntu soieus fsoens fsoensoo  phcua, jnsh ninc ghrtndgna hghsnr 7n . \nkn niaor soieus, gh rchdoeoio pfsoso knc ihmhpntnc ync` sndn pnkn phrduennc soieus shgng, Ninc kotndgna khc`nc 7ό. Knpnt kotueosinc shgn`no 2

Ϗ ( t + [ )+ Ϙ4 Ϗt + Ϙ + 7 ό  Ϗ[ 4 7 ό  [ 4

7 ό 

(7.1)

Ϗ

Knro Kn ro ph phrsn rsndn dnnc nc kont kontns ns koph kophrf rfeh eha a au augu guc` c`nc nc nc nctn tnrn rn ph phro rofk fkhh kn knc c jrhi jrhiuh uhcs cso, o, shaoc``nc knpnt kophrfena phrsndnnc jrhiuhcso ghroiut 2

j 4

 Ϗ   [  7 ό  >

4

(7.?)

Ifcstnc Ifc stnctn tn d47zj d47zj koshgu koshgutt khc`nc khc`nc jrhiuh jrhiuhcso cso fsoenso fsoenso.. Ghsnrnc Ghsnrnc jrhiuhc jrhiuhcso so

fsoenso fsoenso

kocyntninc knend sntunc rnkonc phr shifc. Jrhiuhcso knc phrofkh ghgnc d pnkn shguna ph`ns i ghrinotnc khc`nc ifcstnctn ifcstnctn ph`ns. Npngoen Npngoen dosneinc Ϗ 4

−

 i   , d

dnin koknpntinc ansoe 2 7 ό 

(7.>  i    4 7 ό  7 ό  d

(7.>>)

Knc

j 4

−  −

  i    d

[ 4

7. 7.;. ;.

Ϗ

  47 ό 

Hchr Hchr`o `o Fso Fsoen entf tfrr Anrd Anrdfc fcoi oi ]hk ]hkhr hran ancn cn

Xctui shguna fsoentfr anrdfcom shkhrancn, phr`hshrnc/sodpnc`nc nknena

 x 4 N soc ( Ϗ ¶¶ < t + Ϙ ) ¶ Khc`nc ihmhpntnccyn nknena 2

1

 

 x 4

kx  4 Ϗ<  Nmfs ( Ϗ < t + Ϙ ) kt 

Knc coeno dnisodud ihmhpntnc v 7

7

7

Z   (( x )4  I N soc ( Ϗ < t + Ϙ ) Z  ( ( x )4Z < soc ( Ϗ< t + Ϙ ) 7

(=.=)

Kodncn Z  7

Z  7

 H 4 I + Z 4 dx

7

> +  I x 7 7

Iotn phrfeha sfeuso uctui x dhcbnko

 x 4 Nsoc ( Ϗ < t + Ϙ> )  ntnu  x 4 Nsoc ( Ϗ < t + Ϙ 7)

−

Ϙ > kncϘ7 nknena ifcstnctn, ihtoin ndpeotukh N koghroinc  N 4 7 H  gnawn x i   ghrnkn ko nctnrn N+ knc N- ynotu ghrnkn +

−

7 H

i 

 knc ∗

−

7 H

i 

.

Inrhcn Inrh cn ancyn ancyn khc`nc khc`nc oco coeno coeno x dhcbnk dhcbnkoo rhne shphrto shphrto pnkn pnkn phrsnd phrsndnnc nnc kontns kontns.. Anr`n x ihdukonc threhtni konctnrn kun gntns ync` kothctuinc feha hchr`y H knc ifcstnctn ph`ns i. Xctui dhcknpntinc coeno knro Z knc I uctui sntu phrofkh wnitu ifdpeot, iotn dhc``ucninc hisprhso udud uctui coeno rntn-rntn iunctutns ynotu 2

?

 

[ 

[ 

<

<

∩ Zkt  ∩ ( Ϗ t +Ϙ ) kt  <

( Z  ) )4

4

[ 

∩

kt 

[ 

 

> > 4 Z  I   ))4  H ( Z   ))4 (  I  7

Iotn dhc`aotuc` rntn-rntn runc` uctui sntu phrofkh phcua

( v ) spnmh 4 >  I N 7 , ( I ) spnmh 4 >   I N 7, knc =

:

(H)spnmh 4 (Z)spnmh + (I)spnmh 4 (H)todh

7.8. \hrsndnnc Fsoenso

\nkn `ndgnr 7.;. kotucbuiinc suntu sosthd ph`ns-dnssn ync` koihcno `nyn eu eunr nr aoc` aoc``n `n dhc` dhc`ne nend ndoo fs fsoe oenso nso pn pnis isn. n. @nyn @nyn kf kfrf rfc` c` kn knro ro eu eunr nr ko konsu nsuds dsoi oinc nc koghroinc shvnrn phrofkoi.

@ndgnr 7.;. Fsoenso pnkn sosthd ph`ns-dnssn.

Goen pnkn sosthd ph`ns-dnssn thrshgut ghgnc ghrdnssn d ph`ns dhdpucyno ihiniunc ihini unc khc`nc ifcstnctn ifcstnctn ph`ns i , ghsnr rhkndnc koshiotnr sosthd kocyntninc feha jnitfr jnitfr rhkndnc g, knc `nyn phrfokoi phrfokoi phcyhgng phcyhgng fsoenso  J(t)o, dnin dhcurut dhcurut au auiu iud d ih ihku kunn Chwt Chwtfc fc,, ph phrs rsnd ndnn nnc c `h `hrni rni gh ghgn gnc c shend shendnn fs fsoe oenso nso gh ghre renc nc`s `suc uc` ` kocyntninc knend ghctui2 7

d

 k  x

kx kt 

 + g  + ix 4 J ( t ) 7

kt 

(7.8>)

`nyn eunr phrofkoi J(t) ududcyn knend ghctui juc`so socus knc mfsocus.

><

 

\hrsndnnc `hrni pnkn sosthd fsoenso pnisn oco thrcyntn okhctoi khc`nc  phrsndnnc ync` dhc``ndgnrinc neornc nrus gfeni-gneoi ( O)   O)  knend sosthd WEM ihtoin koauguc`inc khc`nc th`nc`nc sudghr gfeni-gneoi Z(t) Z(t),, ynotu 2

 E

kO  q + WO + 4Z ( t ) kt  M 

(7.87)

ntnu knend ghctui phrsndnnc kojhrhcsone ync` dhcyntninc neornc duntnc q nknena 7

 E

k q

 + W 7

kt 

 kq q  + 4Z ( t ) kt  M 

 O 4 khc`nc

(7.8=)

kq kt   pnkn phrsndnnc (7.87) knc (7.8=) ghsnrnc M nknena inpnsotncso

inpnsotfr ync` ninc kodunto, E nknena ockuitncso knro eoeotnc ync` ko`ucninc knc  W nknena ghsnr andgntnc eostroi ync` ghrjuc`so shgn`no phdgntns nrus eostroi. Ihokhc Iho khctoi toinc nc khc`nc khc`nc phrsnd phrsndnnc nnc (7.8>) (7.8>) thrshgu thrshgutt dhcyhg dhcyhgngi nginc nc dnssn dnssn ghgnc ghgnc d okhcto okh ctoi i khc`nc khc`nc ockuitn ockuitncso cso  E, jnit jnitfr fr rhkndn rhkndnc c g  ok okhc hcto toi i kh khc` c`nc nc an andg dgntn ntnc c  W  W,, ifcstn ifc stnctn ctn ph`ns ph`ns i   ghrshsunonc khc`nc khc`nc inpnsotncso inpnsotncso M , `nyn phrofkoi phrofkoi koeunr koeunr  J(t) okhcto okh ctoi i khc`nc khc`nc Z( Z(t) t),, shkn shknc`i c`inc nc nrus nrus eostroi eostroi  O okhctoi khc`nc ihmhpntnc ghckn kx/kt . Ihkun Ihkun phrsnd phrsndnnc nnc knc kojhrhc kojhrhcson sonee knro knro ihkun ihkun prfshs prfshs ync` ync` ghrghk ghrghknn oco dhc`ansoeinc phcyhehsnonc khc`nc inrnithrostoi ync` okhctoi.

Mfctfa sfne Boin suntu gfen ghso ync` ghrntcyn w4 1?,